β ~ N(β, σ2/Sr.) and In normal regression analysis, the distributions of the estimators β and...
Let X1, X, be iid M μ σ2). Then, find the joint distributions of (i) 2, , Y, where Y-X,-X,, i = 2, , n; Hint: Use the Definition 4.6.1 for the multivariate normality. FYI: 1) Definition 4.6.1 Ά p(2 1) random vector X-(X1, X is said to have a p-dimensional normal distribution, denoted by N, if and only if each linear function X^ajX, has the univariate normal distribution for all fixed, but arbitrary real numbers a, a,
1. Recall that in the normal linear regression analysis formalism the random variable Y given X- is assumed to follow N(a + ßr, σ2) so that a maximu in likelihood calculation leads to the point estimator -1 (Xtlevarian ee", whi, in.ต),.. Ja m) amdata pairs oll etedforth" pai,ofran km.ariales (X,Y) (a) Randomize σ2 to get a randorn variable Σ2 such that its value is σ2 with the data given. (b) Show that Σ2 obtained in (a) is not an unbiased...
Construct the likelihood function L(0,A, σ 2) of: exp Where Yi-NG, + βί Xi, σ2) and estimate βο, βι and σ2 in Y-β0 + Axi + εϊ , where εί-N(0, σ2) ,using the MLE. Compare the least squares estimators with the MLE. Construct the likelihood function L(0,A, σ 2) of: exp Where Yi-NG, + βί Xi, σ2) and estimate βο, βι and σ2 in Y-β0 + Axi + εϊ , where εί-N(0, σ2) ,using the MLE. Compare the least squares...
Consider the regression model where the εi are i.i.d. N(0,σ2) random variables, for i = 1, 2, . . . , n. (a) (4 points) Show βˆ is normally distributed with mean β and variance σ2 . 1 1SXX Question 6 Consider the regression model y = Bo + B12 + 8 where the €, are i.i.d. N(0,0%) random variables, for i = 1,2, ..., n. (a) (4 points) Show B1 is normally distributed with mean B1 and variances
In the simple linear regression with zero-constant item for (xi , yi) where i = 1, 2, · · · , n, Yi = βxi + i where {i} n i=1 are i.i.d. N(0, σ2 ). (a) Derive the normal equation that the LS estimator, βˆ, satisfies. (b) Show that the LS estimator of β is given by βˆ = Pn i=1 P xiYi n i=1 x 2 i . (c) Show that E(βˆ) = β, V ar(βˆ) = σ...
5. So far in our linear modeling, we have assumed that Ylz ~ NA,+Az,σ2); that is, there is a normal distribution of common variance around the regression line. Here, we change this up! Suppose that X~Unif(0, 1) and that for a given r, we know YlN(,22). (Here, the regression lne is 01z and the variance around the regression grows as r grows.) a. In R, figure out how to generate 1000 data points that follow this model and plot them....
Exercise 2.6: Consider the models y Xßte and y* X"β+c" where E(e) = 0, cov(e) = σ21, y* = ГУ, X* = ГХ, e* =「ε and r is a known n x n orthogonal matrix. Show that: 1. E(e) 0, cov(e) σ21 2. b b and s2 s2, where b and b' are the least squares estimates of β and 82 and s+2 are the estimates of σ2 obtained from the two models.
Correlation and Regression a) Determine the value of a (as a function of n and y), which minimizes S function given below. Prove your answer with all the details. s=30,-a)? b) Consider the following two models: Modell:Y, = B. +B,X, + u Model2:Y,= 2, + QX: +e; where Ñ =X; -X (i) Are the OLS estimators of constant terms for both models identical? Are their variances the same? Prove your answer. Are the OLS estimators of slope terms for both...
Question 2 (10 points) You are given the following model y-put ei. Consider two alternative estimators of β, b2xvix? and b = Zy/X 1. Which estimator would you choose and why if the model satisfies all the assumptions of classical regression? Prove your results. (4 points) 2. Now suppose that var(y)-hxi, where h is a positive constant (a) Obtain the correct variance of the OLS estimator. (2 points) (b) Show that the BLU estimator is now 6. Derive its variance....
If X ~ N(0, σ2), then Y function of Y is X follows a half-normal distribution; i.e., the probability density This population level model might arise, for example, if X measures some type of zero-mean difference (e.g., predicted outcome from actual outcome) and we are interested in absolute differences. Suppose that Yi, ½, ,y, is an iid sample from fy(ylơ2) (a) Derive the uniformly most powerful (UMP) level α test of 2 2 0 versus Identify all critical values associated...